# free product

## Definition

Let $G$ be a group, and let ${({A}_{i})}_{i\in I}$ be
a family of subgroups^{} (http://planetmath.org/Subgroup) of $G$.
Then $G$ is said to be a *free product ^{}* of the subgroups ${A}_{i}$
if given any group $H$ and
a homomorphism

^{}(http://planetmath.org/GroupHomomorphism) ${f}_{i}:{A}_{i}\to H$ for each $i\in I$, there is a unique homomorphism $f:G\to H$ such that ${f|}_{{A}_{i}}={f}_{i}$ for all $i\in I$. The subgroups ${A}_{i}$ are then called the

*free factors*of $G$.

If $G$ is the free product of ${({A}_{i})}_{i\in I}$,
and ${({K}_{i})}_{i\in I}$ is a family of groups such that ${K}_{i}\cong {A}_{i}$
for each $i\in I$,
then we may also say that $G$ is the free product of ${({K}_{i})}_{i\in I}$.
With this definition, every family of groups has a free product,
and the free product is unique up to isomorphism^{}.

The free product is the coproduct^{} in the category of groups.

## Construction

Free groups^{} are simply the free products of infinite cyclic groups,
and it is possible to generalize the construction given in the free group
article to the case of arbitrary free products.
But we will instead construct
the free product as a quotient (http://planetmath.org/QuotientGroup) of a free group.

Let ${({K}_{i})}_{i\in I}$ be a family of groups.
For each $i\in I$,
let ${X}_{i}$ be a set and ${\gamma}_{i}:{X}_{i}\to {K}_{i}$ a function
such that ${\gamma}_{i}({X}_{i})$ generates ${K}_{i}$.
The ${X}_{i}$ should be chosen to be pairwise disjoint;
for example, we could take ${X}_{i}={K}_{i}\times \{i\}$,
and let ${\gamma}_{i}$ be the obvious bijection.
Let $F$ be a free group freely generated by ${\bigcup}_{i\in I}{X}_{i}$.
For each $i\in I$,
the subgroup $\u27e8{X}_{i}\u27e9$ of $F$ is freely generated by ${X}_{i}$,
so there is a homomorphism ${\varphi}_{i}:\u27e8{X}_{i}\u27e9\to {K}_{i}$
extending ${\gamma}_{i}$.
Let $N$ be the normal closure^{} of ${\bigcup}_{i\in I}\mathrm{ker}{\varphi}_{i}$ in $F$.

Then it can be shown that $F/N$ is the free product of the family of subgroups ${(\u27e8{X}_{i}\u27e9N/N)}_{i\in I}$, and ${K}_{i}\cong \u27e8{X}_{i}\u27e9N/N$ for each $i\in I$.

Title | free product |
---|---|

Canonical name | FreeProduct |

Date of creation | 2013-03-22 14:53:34 |

Last modified on | 2013-03-22 14:53:34 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20E06 |

Related topic | FreeProductWithAmalgamatedSubgroup |

Related topic | FreeGroup |

Defines | free factor |